Set resting length for a generic spring constraint
The spring constraint is anchored by the Empty that it lives on.  You may be forced to start the objects in the pre-gravity position, and then animate them to move from that position to the stretched rest position in frame 2.  Then you can deactivate the Animated checkbox with another keyframe and let the simulation take over.

[editing to address questions in the comments:]

The Animated checkbox can be found in the Rigid Body subsection of the Physics section of the Properties view for the object.  The Animated checkbox can have keyframes (so you can give it a keyframe at frame=1 to start it out checked, and then give it another keyframe to uncheck it whenever you need it to switch over to physics simulation).

As for the multi-spring scenario, you may have an incontrovertible point.  I consider it possible to construct a set of spring linkages which could have no zero-tension rest position, and can only be expressed using a baseline tension parameter which blender appears to lack.

Although now that I think about it more, I am imagining a scenario where the spring tensions are set for empties that start out unattached, but then are bound to the target object later in the animation (using the Enabled checkbox in the rigid body constraint in the physics section).  My physics demo catalog contains several examples where constraints are temporarily disabled.  This would still require considerable ingenuity and a non-trivial amount of math to arrange.